Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{-2z^2 + 24z - 70}{7z^2 - 112z + 441}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {-2(z^2 - 12z + 35)} {7(z^2 - 16z + 63)} $ $ a = -\dfrac{2}{7} \cdot \dfrac{z^2 - 12z + 35}{z^2 - 16z + 63} $ Next factor the numerator and denominator. $ a = - \dfrac{2}{7} \cdot \dfrac{(z - 7)(z - 5)}{(z - 7)(z - 9)}$ Assuming $z \neq 7$ , we can cancel the $z - 7$ $ a = - \dfrac{2}{7} \cdot \dfrac{z - 5}{z - 9}$ Therefore: $ a = \dfrac{ -2(z - 5)}{ 7(z - 9)}$, $z \neq 7$